PASS Sample Size Software NCSS.com Chapter 544 One-Way Analysis of Variance Contrasts Introduction The one-way (multiple group) design allows the means of two or more populations (groups) to be compared to determine if at least one mean is different from the others. The F test is used to determine statistical significance. The usual F-test tests the hypothesis that all means are equal versus the alternative that at least one mean is different from the rest. Often, a more specific alternative is desired. For example, you might want to test whether the treatment means are different from the control mean, the low dose is different from the high dose, a linear trend exists across dose levels, and so on. These questions are tested using specific contrasts. A comparison is a weighted average of the means, in which the weights may be negative. When the weights sum to zero, the comparison is called a contrast. PASS provides results for contrasts. To specify a contrast, we need only specify the weights. For example, suppose an experiment conducted to study a drug will have three dose levels: none (control), 20 mg, and 40 mg. The first question is whether the drug made a difference. If it did, the average response for the two groups receiving the drug should be different from the control. If we label the group means M0, M2, and M4, we are interested in comparing M0 with M2 and M4. This can be done in two ways. One way is to construct two tests, one comparing M0 and M2 and the other comparing M0 and M4. Another method is to perform one test comparing M0 with the average of M2 and M4. These tests are conducted using contrasts. The coefficients are as follows: M0 vs. M2 To compare M0 versus M2, use the coefficients -1, 1, 0. When applied to the group means, these coefficients result in the comparison M0(-1) + M2(1) + M4(0) which reduces to M2-M0. That is, this contrast results in the difference between two group means. We can test whether this difference is non-zero using the t test (or F test since the square of the t test is an F test). M0 vs. M4 To compare M0 versus M4, use the coefficients -1, 0, 1. When applied to the group means, these coefficients result in the comparison M0(-1) + M2(0) + M4(1) which reduces to M4 - M0. That is, this contrast results in the difference between the two group means. M0 vs. Average of M2 and M4 To compare M0 versus the average of M2 and M4, use the coefficients -2, 1, 1. When applied to the group means, these coefficients result in the comparison M0(-2) + M2(1) + M4(1) which is equivalent to M4 + M2 - 2(M0). 544-1 © NCSS, LLC. All Rights Reserved. PASS Sample Size Software NCSS.com One-Way Analysis of Variance Contrasts Assumptions Using the F test requires certain assumptions. One reason for the popularity of the F test is its robustness in the face of assumption violation. However, if an assumption is not even approximately met, the significance levels and the power of the F test are invalidated. Unfortunately, in practice it often happens that several assumptions are not met. This makes matters even worse. Hence, steps should be taken to check the assumptions before important decisions are made. The assumptions of the one-way analysis of variance are: 1. The data are continuous (not discrete). 2. The data follow the normal probability distribution. Each group is normally distributed about the group mean. 3. The variances within the groups are equal. 4. The groups are independent. There is no relationship among the individuals in one group as compared to another. 5. Each group is a simple random sample from its population. Each individual in the population has an equal probability of being selected in the sample. Technical Details for One-Way ANOVA Contrasts Suppose G groups each have a normal distribution and equal means ( = = = ). Let = = = denote the number of subjects in each group and let N denote the total sample size of all groups. Let denote the weighted mean of all groups. That is 1 2 ⋯ 1 2 ⋯ = � � � Let denote the common standard deviation of all groups.=1 Suppose you want to test whether the contrast C = � is significantly different from zero. Here the ci’s are the contrast=1 coefficients. Define = / 2 �� � � � =1 =1 Define the noncentrality parameter λC, as = / 2 2 544-2 © NCSS, LLC. All Rights Reserved. PASS Sample Size Software NCSS.com One-Way Analysis of Variance Contrasts Power Calculations for Contrasts The calculation of the power of a test proceeds as follows: 1. Determine the critical value, F1,N-G,α, where α is the probability of a type-I error and G and N are defined above. Note that this is a two-tailed test as no direction is assigned in the alternative hypothesis. 2. From a hypothesized set of μi’s, calculate the noncentrality parameter λC. 3. Compute the power as the probability of being greater than F1,N-G,α on a noncentral-F distribution with noncentrality parameter λC. Procedure Options This section describes the options that are specific to this procedure. These are located on the Design tab. For more information about the options of other tabs, go to the Procedure Window chapter. Design Tab The Design tab contains most of the parameters and options that you will be concerned with. Solve For Solve For This option specifies the parameter to be solved for from the other parameters. Select either Power for a power analysis or Sample Size for a sample size determination. Power and Alpha Power This option specifies one or more values for power. Power is the probability of rejecting a false null hypothesis, and is equal to one minus Beta. Beta is the probability of a type-II error, which occurs when a false null hypothesis is not rejected. In this procedure, a type-II error occurs when you fail to reject the null hypothesis of a zero contrast value when in fact the contrast value is not zero. Values must be between zero and one. Historically, the value of 0.80 (Beta = 0.20) was used for power. Now, 0.90 (Beta = 0.10) is also commonly used. A single value may be entered here or a range of values such as 0.8 to 0.95 by 0.05 may be entered. Alpha This option specifies one or more values for the probability of a type-I error. A type-I error occurs when a true null hypothesis is rejected. In this procedure, a type-I error occurs when you reject the null hypothesis of a zero contrast value when the contrast value is zero. Values must be between zero and one. Historically, the value of 0.05 has been used for alpha. This means that about one test in twenty will falsely reject the null hypothesis. You should pick a value for alpha that represents the risk of a type-I error you are willing to take in your experimental situation. You may enter a range of values such as 0.01 0.05 0.10 or 0.01 to 0.10 by 0.01. 544-3 © NCSS, LLC. All Rights Reserved. PASS Sample Size Software NCSS.com One-Way Analysis of Variance Contrasts Sample Size and Group Allocation G (Number of Groups) This is the number of groups (arms) whose means are being compared. The number of items used in the Group Allocation boxes is controlled by this number. This value must be an integer greater than or equal to two. Group Allocation Input Type (when Solve For = Power or Effect Size) Specify how you want to enter the information about how the subjects are allocated to each of the G groups. Possible options are: • Equal (N1 = N2 = ... = NG) The sample size of all groups is Ni. Enter one or more values for the common group sample size. • Enter group multipliers Enter a list of group multipliers (r1, r2, ..., rG) and one or more values of Ni. The individual group sample sizes are found by multiplying the multipliers by Ni. For example, N1 = r1 x Ni. • Enter N1, N2, ..., NG Enter a list of group sample sizes, one for each group. • Enter columns of Ni's Select one or more columns of the spreadsheet that each contain a set of group sample sizes going down the column. Each column is analyzed separately. Ni (Subjects Per Group) Enter Ni, the number of subjects in each group. The total sample size, N, is equal to Ni x G. You can specify a single value or a list. Single Value Enter a value for the individual sample size of all groups. If you enter '10' here and there are five groups, then each group will be assigned 10 subjects and the total sample size will be 50. List of Values A separate power analysis is calculated for each value of Ni in the list. All analyses assume that the common, group sample size is Ni. Range of Ni Ni > 1 Group Multipliers (r1, r2, ..., rG) Enter a set of G multipliers, one for each group. The individual group sample sizes is computed as Ng = ceiling[rg x Ni], where ceiling[y] is the first integer greater than or equal to y. For example, the multipliers {1, 1, 2, 2.95} and base Ni of 10 would result in the sample sizes {10, 10, 20, 30}. Incomplete List If the number of items in the list is less than G, the missing multipliers are set equal to the last entry in the list.